The Permanent Functions of Tensors
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The Permanent Functions of Tensors Qing-Wen Wang1 · Fuzhen Zhang2
Received: 8 December 2017 / Revised: 13 February 2018 / Accepted: 19 March 2018 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018
Abstract By a tensor we mean a multidimensional array (matrix) or hypermatrix over a number field. This article aims to set an account of the studies on the permanent functions of tensors. We formulate the definitions of 1-permanent, 2-permanent, and k-permanent of a tensor in terms of hyperplanes, planes, and k-planes of the tensor; we discuss the polytopes of stochastic tensors; at the end, we present an extension of the generalized matrix function for tensors. Keywords Birkhoff-von Neumann theorem · Doubly stochastic matrix · Hypermatrix · Matrix of higher order · Multidimensional array · Permanent · Polytope · Stochastic tensor · Tensor Mathematics Subject Classification (2010) 15A15 · 15A02 · 52B12
1 Introduction The study on multidimensional arrays (or matrices) may date back as early as the nineteenth century by Cayley [7, 8]. Jurkat and Ryser revived the topic in their seminal paper [21] in 1968 in which they investigated configurations and decompositions for multidimensional arrays. Jurkat and Ryser’s work was followed by a great deal of research on the topic, mainly on the combinatorial aspects of certain types (such as stochasticity) of multidimensional
Fuzhen Zhang
[email protected] Qing-Wen Wang [email protected] 1
Shanghai University, Shanghai, People’s Republic of China
2
Nova Southeastern University, Fort Lauderdale, FL, USA
Q.-W. Wang, F. Zhang
arrays (see, e.g., Brualdi and Csima [3, 5]). In recent years, multidimensional arrays found applications in practical fields such as image processing (see, e.g., Qi and Luo [32]), theory of computing (see, e.g., Cifuentes and Parrilo [12]), and physics (see, e.g., Tichy [36]). We are concerned with the permanent functions of multidimensional arrays. Our purpose is to set an account on the specific topic based on publications, including, in particular, the ones by Dow and Gibson [16] and Taranenko [35]. The results are expositorily presented with explanations other than in the format of theorem-proofs. Some results are easy observations; they are not necessarily new. For the determinants of multidimensional arrays, hyperdeterminants, and related topics, see, e.g., [19, 20, 26, 34]. Let n1 , n2 , . . . , nd be positive integers. We write A = (ai1 i2 ...id ), ik = 1, 2, . . . , nk , k = 1, 2, . . . , d, for an n1 × n2 × · · · × nd multidimensional array or hypermatrix of order d (the number of indices). Multidimensional arrays, or hypermatrices, or matrices of higher orders, are referred to as tensors (see, e.g., [15, 24, 32]). So, by a tensor we mean a multidimensional array. The tensors of order 1 (i.e., d = 1) are vectors in Rn1 , while the second-order tensors are just regular n1 × n2 matrices. A third-order tensor, i.e., an n1 × n2 × n3 tensor, may be viewed as a book of n3 pages (sli
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